On Coxeter Type Classification of Loop-Free Edge-Bipartite Graphs and Matrix Morsifications

We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11 and SYNASC12) of the root systems in the sense of Bourbaki, the mesh geometries Γ(R_Δ, Φ_A) of roots of Δ in the sense of [J. Pure Appl. Algebra, 215 (2010), 13-34], and matrix morsifications A ∈ Mor_Δ, for simply laced Dynkin diagrams Δ ∈ {A_n, D_n, E₆, E₇, E₈}. Here we report on algorithmic and morsification technique for the Coxeter spectral analysis of connected loop-free edge-bipartite graphs Δ with n ≥ 2 vertices by means of the Coxeter matrix Cox_Δ ∈ M_n(Z), the Coxeter spectrum specc_Δ, and an inflation algorithm associating to any connected loop-free positive bigraph Δ a simply laced Dynkin diagram DΔ, and defining a Z-congruence of the symmetric Gram matrices G_Δ and G_DΔ. We also present a computer aided technique that allows us to construct a Z-congruence of the non-symmetric Gram matrices Ğ_Δ and Ğ_Δ', if the Coxeter spectra specc_Δ and specc_Δ' coincide. A complete Coxeter spectral classification of positive edge-bipartite graphs Δ of Coxeter-Dynkin types DΔ ∈ {A_n, D_n, E₆, E₇}, with n ≤ 7, is obtained by a reduction to computer calculation of Gl(n, Z)_DΔ-orbits in the set Mor_DΔ, where Gl(n, Z)_DΔ is the isotropy group of the Dynkin diagram DΔ.